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gather, is to form a Theory of Systems of Rays. Finally, to do
this in such a manner as to make available the powers of the
modern Diathesis, replacing figures by functions and diagrams
by formulae, is to construct an Algebraic Theory of such
Systems, or an Application of Algebra to Optics.
'Towards constructing such an application it is natural, or
rather necessary, to employ the method introduced by Des-
cartes for the application of Algebra to Geometry. That great
and philosophical mathematician conceived the possibility, and
employed the plan, of representing or expressing algebraically
the position of any point in space by three co-ordinate numbers
which answer respectively how far the point is in three rectan-
gular directions (such as north, east, and west), from some fixed
point or origin selected or assumed for the purpose; the three
dimensions of space thus receiving their three algebraical
equivalents, their appropriate conceptions and symbols in the
general science of progression [order]. A plane or curved surface
became thus algebraically defined by assigning as its equation
the relation connecting the three co-ordinates of any point upon
it, and common to all those points: and a line, straight or
curved, was expressed according to the same method, by the
assigning two such relations, correspondent to two surfaces of
which the line might be regarded as the intersection. In this
manner it became possible to conduct general investigations
respecting surfaces and curves, and to discover properties
common to all, through the medium of general investigations
respecting equations between three variable numbers: every
geometrical problem could be at least algebraically expressed,
if not at once resolved, and every improvement or discovery in
Algebra became susceptible of application or interpretation in
Geometry. The sciences of Space and Time (to adopt here a view
of Algebra which I have elsewhere ventured to propose) became
intimately intertwined and indissolubly connected with each
other. Henceforth it was almost impossible to improve either
science without improving the other also. The problem of
drawing tangents to curves led to the discovery of Fluxions or
Differentials: those of rectification and quadrature to the inver-
sion of Fluents or Integrals: the investigation of curvatures of